A short interval result for the e-squarefree e-divisor function.

[section] 1. Introduction and preliminaries

Let n > 1 be an integer of canonical from n = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of n if [b.sub.i]|[a.sub.i] for every i [member of]{1,2, ..., s}, notation: d|[sub.e]n. By convention 1|[sub.e]1.

The integer n > 1 is called e-squarefree, if all exponents [a.sub.1], ..., [a.sub.s] are squarefree. The integer 1 is also considered to be e-squarefree. Consider now the exponential squarefree exponential divisor (e-squarefree e-divisor) of n. Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an e-squarefree e-divisor of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are squarefree. Note that the integer 1 is e-squarefree but is not an e-divisor of n > 1.

Let [t.sup.(e)](n) denote the number of e-squarefree e-divisor of n. The function [t.sup.(e)](n) is called the e-squarefree e-divisor function, which is a multiplicative and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then (see [1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [omega]([alpha]) = s denotes the number of distinct prime factors of [alpha]. The properties of the function [t.sup.(e)](n) were investigated by many authors, see example [4]. Let

A(x) : = [summation over (n[less than or equal to]x])] [t.sup.(e)](n),

Recently Laszlo Toth proved that the estimate

[summation over (n[less than or equal to]x)] [t.sup.(e)](n) = [c.sub.1]x + [c.sub.2][x.sup.1/2] + O([x.sup.1/4 + [epsilon]])

holds for every [epsilon] > 0, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (*)

Throughout this paper, [epsilon] always denotes a fixed but sufficiently small positive constant. We assume that 1 [less than or equal to] a [less than or equal to] b are fixed integers, and we denote by d(a, b; k) the number of representations of k as k = [n.sup.a.sub.1][n.sup.b.sub.2], where [n.sup.1], [n.sup.2] are natural numbers, that is,

d(a, b; k) = [summation over (k = [n.sup.a.sub.1][n.sup.b.sub.2])] 1,

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be used freely.

The aim of this short text is to study the short interval case and prove the following.

Theorem. If [x.sup.1/5+2[epsilon]] < y [less than or equal to] x, then

[summation over (x < n [less than or equal to] x + y])] [t.sup.(e)](n) = [c.sub.1]y + O([yx.sup.-[epsilon]/2 + [x.sup.1/5+3/2[epsilon]]),

where [c.sub.1] is given by (*).

[section] 2. Proof of the theorem

In order to prove our theorem, we need the following lemmas.

Lemma 1. Suppose s is a complex number (Res > 1), then

F(s) := [[infinity].summation over (n=1)] [t.sup.(e)](n)/[n.sup.2] = [zeta](s)[zeta](2s)/[zeta](4s) G(s),

where the Dirichlet series G(s) := [[summation].sup.[infinity].sub.n=1] g(n)/[n.sup.s] is absolutely convergent for Res > 1/6.

Proof. Here [t.sup.(e)] (n) is multiplicative and by Euler product formula we have for [sigma] > 1 that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

So we get G(s) := [[summation].sup.[infinity].sub.n=1] g(n)/[n.sup.s]. It is easily seen the Dirichlet series is absolutely convergent for Res > 1 /6.

Lemma 2. Let k [greater than or equal to] 2 be a fixed integer, 1 < y [less than or equal to] x be large real numbers and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have

B(x, y; k, [epsilon]) [much less than] [yx.sup.-[epsilon]] + x 1/[x.sup.2k+1] log x. (2)

Proof. This lemma is very important when studying the short interval distribution of 1-free number, see example [3].

Lemma 3. Let a(n) be an arithmetic function defined by (2), then we have

[summation over (n[less than or equal to]x)] a(n) = Cx + O([x.sup.1/6] + [epsilon]), (3)

where C = [Res.sub.s=1][zeta](s)G(s).

Proof. Using Lemma 1, it is easy to see that

[summation over (n[less than or equal to]x)[absolute value of g(n)] [much less than] [x.sup.1/6+[epsilon]].

Therefore from the definition of g(n) and (2), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and C = [Res.sub.s=1][zeta](s)G(s).

Next we prove our theorem. From Lemma 3 and the definition of a(n), we get

[t.sup.(e)](n) = [summation over (n=[n.sub.1][n.sup.2.sub.2][n.sup.4.sub.3])] a([n.sub.1])[mu]([n.sup.3]),

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

So we have

A(x + y) - A(x) = [summation over (x=[n.sub.1][n.sup.2.sub.2][n.sup.4.sub.3][less than or equal to]x+y)] a([n.sub.1])[mu]([n.sub.3]) = [summation over (1)] + O([summation over (2)] + [summation over (3)]). (5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

In view of Lemma 3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [c.sub.1] = [Res.sub.s=1] F(s).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Similarly we have

[summation over (3)] [much less than] [yx.sup.[epsilon]/2] + [x.sup.1/5] + 3/2 [epsilon]]. (9)

Now our theorem follows from (5)-(9).

References

[1] Laszlo Toth, An order result for the exponential divisor function, Publ. Math. Debrean, 71(2007), No. 1-2, 165-171.

[2] Laszla Toth, On certain arithmetic function involing exponential divisors, II. Annales Univ. Sci. Budapest. Sect. Comp., 27(2007), 155-156.

[3] M. Filaseta, O. Trifonov, The distribution of square full numbers in short intervals, Acta Arith., 67(1994), No. 4, 323-333.

[4] Heng Liu and Yanru Dong, On the mean value of the e-squarefree e-divisor function. Scientia Magna, 5(2009), No. 4, 46-50.

[5] M. Berkani, On a class of quasiFredholm operators, Integral Equations Operator Theory, 34(1999), 244-249.

Mengluan Sang ([dagger]), Wenli Chen ([double dagger]) and Yu Huang (#)

([dagger])([double dagger]) School of mathematical Sciences, Shandong Normal University, Jinan, 250014

(#) Network and Information Center, Shandong University, Jinan, 250100

E-mail: [e-mail] [e-mail] [e-mail]

(1) This work is supported by N. N. S. F. of China (Grant Nos: 10771127, 11001154) and N. S. F. of Shandong Province (Nos: BS2009SF018, ZR2010AQ009).